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64 = 65? Frih$ reward offered! Please help.





uuuuuu
Please take a look at This Site.

I am looking at this site, and it magically turns an area of 64 to an area of 65.
This is not possible, is it?

They say it is a trick, but I look at it, and I count the sizes, and it is clearly true. 64 really does equal 65.

I am so bewildered by this that I can not get any sleep tonight. It is all I can think about.

If it is acceptable to the staff of FriHost, I am willing to offer a reward of Frih$ to the person who can convince me that 64 is not actually the same as 65, and that the world is not falling apart.

Please help me sleep again!

Love,

Uwe
Pilot
I think it has something to do with some of the inside angle edges being put to the outside, creating more surface area. But that's about as deep as this simple brain can dig into this problem... hope I've got someone else started.
Aless
It's just a mind-trick in how they cut it. The line going through the rectangle creates the illusion of extra space, thus making you think you're seeing 65 in stead of the original 64. Stuff like this pops up on the internet all the time.
Arnie
http://mathsucks.ytmnd.com ==> http://mathrules.ytmnd.com
Does that answer your question?
silvermesh
since the ytmnd rebuttal doesn't actually try to prove anything other than drawing extra white space, it should be pointed out here that the angles on the two cuts don't fit together to form a straight line. The person who did the ytmnd rebuttal made this more obvious by adding extra space in between, and the person who made this applet doing the original faulty dissection experiment is trying to teach you not to trust that what looks to be tru is always true. the pieces are overlapping(and if you go to his other example, he points out how he couldn't quite make it look right in the applet due to the overlap). the pieces don't actually fit together the way they are displayed to be, thus eliminating your paradox.

as far as FRIH$ are concerned, you don't even need a reason to give them to someone. What you do with your FRIH$ is your own decision.
uuuuuu
Thanks to all concerned for the kind assistance.
Unfortunately for me, I may be so slow as to be beyond help.
Does the logic bomb illustrate that the geometry of spacetime has been altered?
And if so, what is the implication of that?
JJGY
uuuuuu wrote:
Thanks to all concerned for the kind assistance.
Unfortunately for me, I may be so slow as to be beyond help.
Does the logic bomb illustrate that the geometry of spacetime has been altered?
And if so, what is the implication of that?

The so called logic bomb simply tries to illustrate the belief of the author that the trick is really nothing other than a simple overlapping of the pieces.
NuniPio
the green and blue overlap a bit in the 8x8 block, the overlap space = 1 block
Davidgr1200
Just take a look at http://www.cut-the-knot.org/Curriculum/Geometry/FibonacciBamboozlement.shtml which the site refers to
raine dragon
the pieces don't actually fit properly. There is a small gap between them. In the site you showed, the pieces are cut improperly so they overlap, and make it seem as though they fit exactly, but in reality, there is one square of space which takes place in the space between the shapes ^_^
bogger
Actually, it's a lot simpler, in the bottom right corner, there's a number, put that down to 3 and it gets a lot more obvious

If you drag the green one over, it'll grow a bit on it's topmost side

It goes from 1 square wide at the top and 2 at the bottom to 1 and a third at the top and 2 at the bottom

case closed, they cheated
uuuuuu
Thanks everybody!

I get now that the square disappears because of cutting the paper with the scissors and some falls off.

The joke is on me!

Love,

Uwe
Billwaa
lol, I would never thought of that. This is kind of interesting, maybe I should show my Geometry Teacher, see what he have to say about that, lol
NuniPio
so who gets the Frih$??
smartpandian
hey... that's a good link.. obviously one can find thousands of links like this.. may be I can work something like this in my webspace..

So this game is not fair...

So to do the justice, your 64 frih has to be distributed to all Frihosters.. equally..
NuniPio
NuniPio wrote:
the green and blue overlap a bit in the 8x8 block, the overlap space = 1 block


that was my answer, i should get the FRIH$
uuuuuu
Each of the answers that I have seen so far have provided insight toward the missing square problem.

Here is my own answer -- it is neither complete, nor mathematically rigorous, but it addresses the main problem at hand, from my perspective:

The original link connected with a java applet, which creates an illusion that if triangles and quadrilaterals with a total area of 64 units are rotated and translated, their area is not preserved. By some magic, it becomes 65 units, and to support this magic, the pieces which once fit perfectly into an 8x8 grid(area = 64) now fit perfectly into a 5x13 grid(area = 65).

Well there is some trickery going on here. The mere moving of the polygons can't increase or decrease their size. We know this fact to be true because we move things every day, and under normal circumstances this just doesn't happen.

So everyone who responded, who tried to identify the trick, identified it pretty well. I hope you had fun thinking about this trick.

My explanation appeals to straight-forward geometry.
If you refer to the 5x13 rectangle, you'll see a line drawn down its diagonal, ascending from left to right. Notice this line appears to intersect the points (0,0), (2,5), (3,Cool, and (8,13).
This is one of the sides of every single polygon in the set.

But one of the properties of a line is that its slope can be described by the difference in y of any two points on the line, divided by the difference in x of those two points. The familiar formula is m = (y_2 - y_1)/(x_2 - x_1). We should be able to plug in the x/y values from any two of the points that the line goes through and arrive at the same slope.

But this does not happen. The slope of the line between the points (0,0) and (2,5) is (2-0)/(5-0) or 2/5. The slope of the line between the points (0,0) and (3,Cool is (3-0)/(8-0) or 3/8. 3/8 = 15/40 < 16/40 = 2/5.
So it's impossible for a straight line to go through those three points.

I'm not sure exactly how the applet achieves the illusion, but it could be done by using a parallelogram between the other 4 polygons which has exactly the area of 1, and is hidden beneath the thickness of the line itself. Otherwise the illusion could be achieved by using pentagons instead of triangles, with one interior angle having a measure of less than 1 degree.

This illusion is very convincing, though, because our brains seem to be wired to want to recognize geometric shapes like straight lines.

Thanks to everybody who responded, I wanted people to think about this interesting problem. I will donate Frih$ to everyone who participated if you want it, just post another message on this thread requesting it. You can tell me how much you want, too, and I may or may not be able to accommodate! Smile

Peace to my fellow FriHosters.

-Uwe
justnewbie
bogger wrote:
Actually, it's a lot simpler, in the bottom right corner, there's a number, put that down to 3 and it gets a lot more obvious

If you drag the green one over, it'll grow a bit on it's topmost side

It goes from 1 square wide at the top and 2 at the bottom to 1 and a third at the top and 2 at the bottom

case closed, they cheated


Great answer! I was really got confused by the dissection thing. Have you guys ever heard of a same geometry problem dealing with a triangle?
uuuuuu
I think what you may be talking about is the missing square puzzle. It's very similar to this one.

Here is a link:

http://en.wikipedia.org/wiki/Missing_square_puzzle
JBotAlan
Wow.

That got my mind going for a moment...I would've puzzled on that all night long.

I should present that to my math teacher...she would get a kick out of it! I already got her to say "I HIM WEE TODD DID" (say it yourself...you'll get it)

Thanks for a good puzzle.
JBot
skygaia
At first, it was little bit shocked for me. But as I look aroud it carefully. the small squares are not exactly same. small squares in right picture are little smaller than left.
regards
No way. If this is true, then the whole world should be shocked. This will become a great news. Why?
If 64=65
then 64-1=65-1 (i.e. 63=64)
and goes on...62=63, 61=62......etc
And the numbers on the earth will be the same. There will be no difference between 1 million and 1 cent. Absoulutly, it must be another mathematics trick. Mathematics will automatically shut it down either be definations or proves.

Wink Regards
missdixy
Wow, this totally had me bewildered at first (I clicked the site before reading any of the replies) And once I read all the replies, I felt at peace with the universe again.
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